Assumptions of (a) Set Theory Carl Pollard Ohio State University - - PowerPoint PPT Presentation

Assumptions of (a) Set Theory Carl Pollard Ohio State University Linguistics 680 Formal Foundations Thursday, September 23, 2010

These slides are available at: http://www.ling.osu.edu/∼scott/680

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Sets and Membership

• We assume there exist things called sets.
• We assume there is a relationship, called membership, which,

for any sets A and B, either does or does not hold between them.

• If it does, we say A is a member of B, written A ∈ B.
• If it doesn’t, we say A is not a member of B, written A /

∈ B.

• We never say what sets are: they are the unanalyzed primitives
• f set theory and cannot be reduced to, or explained in terms
• f, more basic things that are not sets.

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Assumptions about the Membership Relation

• We make certain further assumptions about membership.
• Our set theory consists of these additional assumptions plus any

conclusions we can draw from them using valid reasoning.

• For now, we’ll state these assumptions informally in English.
• Later we’ll state them more precisely in a special symbolic lan-

guage (the language of set theory), and the precise restatements

• f the assumptions will be called the axioms of our set theory.
• Also for now we don’t say exactly what counts as valid reasoning.
• Later, we’ll specify what counts as valid reasoning in terms of

mathematical objects called formal proofs which deduce new sen- tences (in the language of set theory) from the axioms.

• There is nothing privileged about our set theory; there are other

set theories which start with different assumptions.

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Assumption 1 (Extensionality) If A and B have the same members, then they are the same set (written A = B).

• We don’t mention that A and B are sets, because we’re doing

set theory (so the only things we are talking about are sets).

• We needn’t assume that if A and B do not have the same mem-

bers, then they are not the same set (written A = B). That’s because, if they were the same set, then everything about them, including what members they had, would be the same.

• If every member of A is a member of B, we say that A is a

subset of B, written A ⊆ B.

• If A ⊆ B, B might have members that are not in A. In that

case we say A is a proper subset of B, written A B.

• But if both A ⊆ B and B ⊆ A, then it follows from Extension-

ality that A = B.

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Assumption 2 (Empty Set) There is a set with no members.

• From this assumption together with Extensionality we can con-

clude that the there is only one set with no members.

• We call this set the empty set, written ‘∅’.
• But later, we’ll sometimes write it as ‘0’.
• That’s because in the usual way of doing arithmetic within set

theory (covered in Chapter 4) zero is the empty set.

• As yet, we have no basis for concluding that there are any sets
• ther than the empty set.
• For example, we are not even able to make a valid argument

that there is a set with ∅ as its only member.

• We remedy this situation by making a few more assumptions.

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Assumption 3 (Pairing) For any sets A and B, there is a set whose members are A and B.

• Even though we say ‘sets’ here, we don’t mean to rule out the

possibility that A and B are the same set.

• Because of Extensionality again, there is only one set whose

members are A and B, which we write as {A, B}, or {B, A}.

• More generally, we will notate any nonempty finite set by listing

its members, separated by commas, between curly brackets, in any order.

• We’ll get clear about what we mean by ‘finite’ in Chapter 5, but

for now we’ll just rely on intuition.

• In listing the members of a set, repititions don’t count, so e.g. if

A and B are the same set, then {A, B} is the same set as {A}.

• So it makes no sense to talk about how many times A is a mem-

ber of B: either it is or it isn’t.

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Definition (Singleton) A singleton is a set with only one member.

• For any set A,we have enough resources now to prove informally

(i.e. make a valid argument in English) that there is a singleton setwhose member is A. (Of course this is written ‘{A}’.

• One singleton set is the set {0} whose member is 0.
• {0} is also written ‘1’, because in the usual way of doing arith-

metic within set theory, it is the same as the number one.

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Preview of the Natural Numbers

• As mentioned above, we’ll define 0 to be ∅, and 1 to be {0}.
• By Pairing, we know there is a set, {0, 1}, whose only members

are 0 and 1. Let’s say that this is what the number 2 is.

• There seems to be a pattern here, in which the next step would

be to say that 3 is the set whose only members are 0, 1, and 2.

• But as yet we don’t have sufficient resources to prove that there

is such a set!

• To say nothing of proving that there is a set whose members are

the natural numbers.

• In fact, as yet we don’t even know what ‘natural number’ means.
• But soon we will (Chapter 4).

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Assumption 4 (Union) For any set A, there is a set whose members are those sets which are members of (at least) one of the members of A.

• Extensionality ensures the uniqueness of such a set, which is

called the union of A, written A.

• If A = {B, C}, then A is the set each of whose members is in

either B or C (or both), usually written B ∪ C.

• In general, B ∪ C is not the same thing as {B, C}!

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Definition (Successor) For any set A, the successor of A, written s(A), is the set A ∪ {A}.

• That is, s(A) is the set with the same members as A, except

that A itself is also a member of s(A).

• Nothing in our set theory will rule out the possibility that A ∈ A,

in which case s(A) = s(A).

• However, some widely used set theories include an assumption

(called Foundation) which does rule out this possibility.

• For example, we can prove that 1 is the successor of 0, and that

2 is the successor of 1.

• Now we can say what 3 is: the successor of 2!
• Likewise we can say what 4, 5, etc. are.
• But we still can’t say exactly what we mean by a natural number.

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Assumption 5 (Powerset) For any set A, there is a set whose members are the subsets of A.

• Again, Extensionality guarantees the uniqueness of such a set,

which we call the powerset of A, written ℘(A).

• In general, ℘(A) is not the same set as A, because usually the

subsets of a set are not the same as the members of the set.

• For example, 0 is a subset of 0 (in fact, every set is a subset
• f itself), but obviously 0 is not a member of 0 (since 0 is the

empty set).

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Assumptions vs. Definitions a. Notice a crucial difference between the successor of a set A and the powerset of A: successor is defined in terms of things whose existence can already be established on the basis of previous assumptions (singletons, unions), whereas the existence of the powerset of A is assumed. b. Why didn’t we just define ℘(A) to be the set whose members are the subsets of A? c. It’s because nobody has found a valid argument (based on just the first four assumptions) that there is such a set! d. More generally, there is no guarantee that, for an arbitrary con- dition on sets P[x], there is a set whose members are all the sets x such that P[x]. e. But this fact did not become known till 1902.

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Russell’s Paradox a. Let P[x] be the condition ‘x is not a member of itself’. b. Following Russell, we will show that there cannot be a set whose members are all the sets x such that P[x]. c. Suppose R were such a set. d. Then either (i) R is a member of itself, or (ii) it isn’t. i. Suppose R is a member of itself, Then it cannot be a member

• f R, since the members of R are sets which are not members
• f themselves. But then it is not a member of itself.

ii. Suppose R is not a member of itself. Then it must be in R. But then, it is a member of itself. iii. Either way, assuming (c) leads to a contradiction. e. So the assumption (c) must have been false.

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An Imaginable Set-Theoretic Assumption Bites the Dust

• Russell’s Paradox shows we don’t have the option of adding the

following to our set theory: Tentative Assumption (Comprehension) For any condition P[x] there is a set whose members are all the sets x such that P[x].

• The following assumption is usually adopted instead.

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Assumption 6 (Separation) For any set A and any condition P[x], there is a set whose members are all the x in A that satisfy P[x].

• So far, assuming Separation has not been shown to lead to a

contradiction.

• Separation is so-called because, intuitively, we are separating
• ut from A some members that are special in some way, and

collecting them together into a set.

• By Extensionality, there can be only one set whose members are

all the sets x in A that satisfy P[x].

• We call that set {x ∈ A | P[x]}.

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Intersection

• In naive introductions to set theory, the intersection of two

sets A and B, written A ∩ B, is often ‘defined’ as the set whose members are those sets which are members of both A and B.

• But how do we know there is such a set?
• If we assume Separation and take P[x] to be the condition x ∈ B,

then we can (unproblematically) define A ∩ B to be {x ∈ A | x ∈ B}.

• A and B are said to intersect provided A ∩ B is nonempty.
• A set A is called pairwise disjoint if no two distinct members
• f it intersect.

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Set Difference

• For two sets A and B, if we take P[x] to be the condition x /

∈ B, then Separation guarantees the existence of the set {x ∈ A | x / ∈ B}.

• This set is called the set difference of A and B, or alternatively

the complement of B relative to A, written A \ B.

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Nonexistence of a Universal Set

• A set is called universal if every set is a member of it.
• We can prove in our set theory that there is no universal set.
• For suppose A were a universal set. Let P[x] be the condition

x / ∈ x. Then by Separation, there must be a set {x ∈ A | x / ∈ x}. But Russell’s argument showed that there can be no such set. So the assumption that there was a universal set must have been false.

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Ordered Pairs

• If A and B are sets, we call the set {{A}, {A, B}} the ordered

pair of A and B, also written A, B.

• A, B differs from {A, B} in the crucial respect that no matter

what A and B are, {A, B} = {B, A}, but A, B = B, A only if A = B.

• More generally, if A, B, C, and D are sets, then A, B = C, D
• nly if A = C and B = D.
• If C is the ordered pair of A and B, A is called the first com-

ponent of C, and B is called the second component of C.

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The Cartesian Product of Two Sets

• For any sets A and B, there is a set whose members are all those

sets which are ordered pairs whose first component is in A and whose second component is in B. (It’s instructive to try to prove

• this. Hint: use Separation.)
• By Extensionality there can be only one such set. It is called

the cartesian product of A and B, written A × B.

• For any sets A, B, C, and D, A × B = C × D only if A = C

and B = D. (Try to prove this.)

• A is called the first factor of A×B, and B the second factor.

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Ordered Triples

• The ordered triple of A, B, and C, written A, B, C, is defined

to be the ordered pair whose first component is A, B and whose second component is C.

• Then A, B, and C are called, respectively, the first, second,

and third components of A, B, C.

• The (threefold) cartesian product of A, B, and C, written

A × B × C, is defined to be (A × B) × C. This is the set of all

• rdered triples whose first, second, and third components are in

A, B, and C respectively.

• The definitions can be extended to ordered quadruples, quin-

tuples, etc., and to n-fold cartesian products for n > 3, in an

• bvious way.

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Definitions (Cartesian Powers) For any set A, a cartesian power of A is a cartesian product all of whose factors are A.

• The first cartesian power of A is just A, also written A(1).
• The cartesian square of A, written A(2), is A × A.
• The cartesian cube of A, written A(3), is A × A × A
• More generally, for n > 3, the n-th cartesian power of A,

written A(n), is the n-fold cartesian product all of whose factors are A.

• Additionally, the zero-th cartesian power of A, A(0), is de-

fined to be the set 1 (= {∅}).

• This last definition is closely related to the arithmetic fact that

for any natural number n, n0 = 1, but we postpone the expla- nation.

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